The exceptional geometry of knots

Joel Fine (Université Libre de Bruxelles)

Monday 19th May 16:00-17:00
Maths 311B

Abstract

I will describe a series of conjectures relating certain topological knot invariants to the following geometric objects: minimal surfaces in hyperbolic 4-space H^4, J-holomorphic curves in H^4xS^2 (which carries an SU(3) structure) and associative submanifolds in H^4xS^2xR (which carries a G2 structure). Given a knot or link L in S^3, the idea is to count minimal surfaces in H^4 which have L as their ideal boundary. The first conjecture is that the number of minimal surfaces should give the coefficients of the HOMFLYPT polynomial of L. These minimal surfaces lift to J-holomorphic curves in H^4xS^2 and so this is really a statement about Gromov-Witten type invariants. The second conjecture says the following: there is a complex generated by these J-holomorphic curves where the differential is given by counting cylindrical associative 3-manifolds in H^4xS^2xR which are asymptotic at each end to a pair of J-holomorphic curves. The conjecture claims that the homology groups of this complex are the Khovanov-Rozansky homology groups of the link L. (These have three gradings and if you take the Euler characteristic over one grading you obtain the coefficients of the HOMFLYPT polynomial.) The talk will be completely non-technical. On the plus side there will be plenty of pictures. On the downside there will be no proofs.

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